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The EM Algorithm for Latent Class Analysis with Equality Constraints

Published online by Cambridge University Press:  01 January 2025

AB Mooijaart  and
Peter G. M. van der Heijden
AB Mooijaart*
Affiliation:
University of Leiden
Peter G. M. van der Heijden
Affiliation:
Department of Empirical and Theoretical Sociology, University of Utrecht
*
Requests for reprints should be sent to Ab Mooijaart, Department of Psychology, Leiden University, Wassenaarseweg 52, 2333 AK, Leiden, THE NETHERLANDS.
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Abstract

The EM algorithm is a popular iterative method for estimating parameters in the latent class model where at each step the unknown parameters can be estimated simply as weighted sums of some latent proportions. The algorithm may also be used when some parameters are constrained to equal given constants or each other. It is shown that in the general case with equality constraints, the EM algorithm is not simple to apply because a nonlinear equation has to be solved. This problem arises, mainly, when equality constraints are defined over probabilities in different combinations of variables and latent classes. A simple condition is given in which, although probabilities in different variable-latent class combinations are constrained to be equal, the EM algorithm is still simple to apply.

Keywords

latent class analysisEM algorithmequality constraints
Type
Original Paper
Information
Psychometrika , Volume 57 , Issue 2 , June 1992 , pp. 261 - 269
Copyright
Copyright © 1992 The Psychometric Society

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Footnotes

The authors are grateful to the Editor and the anonymous reviewers for their helpful comments on an earlier draft of this paper. C. C. Clogg and R. Luijkx are also acknowledged for verifying our results with their computer programs MLLSA and LCAG, respectively.

References

Clogg, C. C. (1977). Unrestricted and restricted maximum likelihood latent structure analysis: A manual for users, University Park, PA: Pennsylvania State University.Google Scholar
Dempster, A. P., Laird, N. M., Rubin, D. B. (1971). Maximum likelihood from incomplete data via the EM algorithm. Journal of the Royal Statistical Society, Series B, 39, 1–38.CrossRefGoogle Scholar
Eliason, S. R. (1988). The categorical data analysis system. Version 3.00A user's manual, University Park, PA: Pennsylvania State University, Department of Sociology.Google Scholar
Formann, A. K. (1978). A note on parameter estimation for Lazarsfeld's latent class analysis. Psychometrika, 43, 123–126.CrossRefGoogle Scholar
Goodman, L. A. (1974). Exploratory latent structure analysis using both identifiable and unidentifiable models. Biometrika, 61, 215–231.CrossRefGoogle Scholar
Goodman, L. A. (1979). A note on the estimation of parameters in latent structure analysis. Psychometrika, 44, 123–128.CrossRefGoogle Scholar
Haberman, S. J. (1979). Analysis of qualitative data, New York: Academic Press.Google Scholar
Haberman, S. J. (1988). A stabilized Newton-Raphson algorithm for loglinear models for frequency tables derived by indirect observation. In Clogg, C. C. (Eds.), Sociological methodology (pp. 193–211). Washington, DC: American Sociological Association.Google Scholar
Hagenaars, J. A., Luijkx, R. (1990). LCAG, Tilburg: Tilburg University, Department of Sociology.Google Scholar
Langeheine, R. (1988). New developments in latent class theory. In Langeheine, R., Rost, J. (Eds.), Latent trait and latent class models (pp. 77–108). New York: Plenum Press.CrossRefGoogle Scholar
van de Pol, F. J. R., Langeheine, R. (1990). Mixed markov latent class models. In Clogg, C. C. (Eds.), Sociological methodology, 1990 (pp. 213–247). Oxford: American Sociological Association.Google Scholar
van de Pol, F. J. R., Langeheine, R., De Jong, W. (1989). PANMARK user manual, Panel analysis using Markov chains, version 1.5, Voorburg: Netherlands Central Bureau of Statistics.Google Scholar